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The Prisoner's Dilemma
A brief description of a simple yet profound problem in game theory.
(The initial posing of the problem was in terms of a pair of prisoners making deals with prosectors in a situation in which each could get a reduced sentence by betraying the other to the authorities. I think it was originally studied as a model for superpower confrontation.)
Now, in a single game of the Dilemma the optimal strategy is clearly to defect. If your opponent cooperates then you can sucker him by defecting and make more points, but if he defects then you can at best equal his score by also defecting. The whole thing doesn't seem very interesting. Things become much more complex when the same pair of players play against each other repeatedly and can modify their behaviour based on the previous rounds. This is the Iterated Prisoner's Dilemma.
In a game of IPD it turns out that the best strategies incorporate the ideas of "niceness" (start by cooperating), "retaliation" (punish those who defect), "forgiveness" (don't get caught up in vendettas - cooperation is better) and "transparency" (randomness is a very bad strategy, and if your strategy is too complex then it behaves pretty much like a random strategy). There are even more complex variants of the game with populations of strategies playing against each other. It would probably be fun to enter an algorithm in an IPD tournament.